On the selection of an optimal outer approximation of a coherent lower probability

Coherent lower probabilities are one of the most general tools within Imprecise Probability Theory, and can be used to model the available information about an unknown or partially known precise probability. In spite of their generality, coherent lower probabilities are sometimes difficult to deal with. For this reason, in previous papers we studied the problem of outer approximating a given coherent lower probability by a more tractable model, such as a 2- or completely monotone lower probability. Unfortunately, such an outer approximation is not unique in general, even if we restrict our attention to those that are undominated by other models from the same family. In this paper, we investigate whether a number of approaches may help in selecting a unique undominated outer approximation. These are based on minimising a distance with respect to the initial model, maximising the specificity, or preserving the same preferences as the original model. We apply them to 2- and completely monotone approximating lower probabilities, and also to the particular cases of possibility measures and p-boxes.